Mahler, Kurt

Reprint: On the approximation of logarithms of algebraic numbers (1953)

Doc. Math. Extra Vol. Mahler Selecta, 527-555 (2019)
DOI: 10.25537/dm.2019.SB-527-555


Mahler gives a new identity by means of which infinitely many algebraic functions approximating the logarithmic function are obtained. On substituting numerical algebraic values for the variable, a lower bound for the distance of its logarithm from variable algebraic numbers is found. As a further application, Mahler proves that the fractional part of the number \(e^a\) is greater than \(a^{-40a}\) for every sufficiently large positive integer \(a\). \par Reprint of the author's paper [Philos. Trans. R. Soc. Lond., Ser. A 245, 371--398 (1953; Zbl 0052.04404)].

Mathematics Subject Classification

11-03, 11J68